A continuous image of a Radon-Nikodym compact which is not Radon-Nikod{y}m

A CONTINUOUS IMAGE OF A RADON-NIKOD ´ YM COMP A CT SP A CE WHICH IS NOT RADO N -NIK OD ´ YM ANTONIO A VIL ´ ES AND PIOTR KOSZMIDER Abstract. W e construct a con tin uous image of a Radon-Nikod´ ym compact space which is not Radon-Nikod´ ym compact, solving the problem posed in the 80ties by Isaac Namiok a. 1. Introduction Recall that a Banach space X has the Radon-Nikod ´ ym pro per ty if and o nly if the Ra don-Nikod´ ym theor em holds for vector measures with v alues in X (see [11]). This prop erty pla ys a cen tral role in the theory of v ector measures. It has been clear for long time that dual Banach spaces with the Rado n-Nikod´ ym pro per ty and their weak ∗ compact subsets play s pec ial role in this theory [31, 3 5]. Isaac Namio k a [24] defined a c ompact spa ce to b e Rado n-Nikod´ ym compact (or RN for short) if and only if it is homeo morphic to a weak ∗ compact subset of a dual Bana ch space with the Radon- Nik o d´ ym pr op erty . F or example, as r eflexive Banach spa ces are dual spaces with the Ra don-Nikod´ ym prop er t y , the r esults of [10] imply that Eb erle in compact spaces are RN compact. Alr eady in [24] a n umber of in teresting prop erties of RN compacta a re proven, as well as an elegant in ternal character ization of RN compacta is giv en. The in v estiga tion of t his class of compact spaces contin ued later, with s ome r emark able results lik e the rela tion with Corson a nd E b erlein co mpact spaces [28, 36]. But the question whic h has attr acted more a tten tion and pro duced a lar ger lit- erature o n RN compacta is the fo llowing very basic problem, already p osed in [24] and tra ced in [14] to [17] which ha s remained o pen up to this date: Is the class of RN c omp act sp ac es close d under c ont inuous images? A num b er of par tial p ositive results to the ab ov e ques tion of contin uous images of RN compacta hav e be en pr ov en. If L is a contin uous image o f an RN compact space, then L is RN compact if an y of the following co nditions hold: (1) L is almost tota lly dis connected [3 ], meaning that L ⊂ [0 , 1] I and for every x ∈ L , |{ i ∈ I : x i ∈ (0 , 1) }| ≤ ω . This includes in particular the c ases when L is zero- dimensional (attributed indep endently to Rez nic henko [2 ]) First author by wa s suppor ted by MEC and FEDER (Pro ject MTM2008-05396), F undaci´ on S ´ eneca (Pro ject 08848/PI/0 8), R amon y Cajal con tract (R YC-2008-02051) and an FP7-PEOPLE- ERG -2008 action. The second author was partially supported b y the National Science Center r esearc h grant DEC-2011/01/B/ST1/ 00657. He also expresses his gratitude to the F unctional Analysis gr oup i n Murcia for constant a nd ongoing support whic h included or ganizing sev eral visits to the Univ ersity of Murcia and made this researc h p ossible. 1 2 ANTONIO A VIL ´ ES AND PIOTR K OSZMIDER and when L is Cor son [3 6], and less o b viously a lso the case when L is linearly ordered [6]. (2) The w eight of L is less than car dinal b [5]. (3) L is the union of tw o RN co mpact subspace s L = L 1 ∪ L 2 and so me s pecia l hypothesis ho ld, like L 1 ∩ L 2 being metrizable, G δ or scatter ed, or whe n L 1 is a retr act of L o r when L \ L 1 is scattered [23]. Other a rticles devoted to the pr oblem of the co nt inuous image include [4, 1 4, 19, 25]. More information ca n b e found in [7, 12, 1 3, 2 6], which are dedica ted to the topic, or contain sections dedicated to it. The purp os e of this article is to provide a neg ative so lution to the general pr oblem: Theorem 1.1. Ther e exists a c ontinuous surje ction π : L 0 − → L 1 such that L 0 is a zer o-dimensional RN c omp act sp ac e but L 1 is not RN c omp act. This co n trasts with other similar classes of compact s paces ar ising in functiona l analysis, like Eb erlein co mpacta (weakly co mpact subsets of Bana ch s paces) or Corson compacta (compa ct subsets of Σ-pr o ducts), for which the stability under contin uous images happe ned to b e a non trivial fact, but was finally shown to hold true in [8] a nd [18] res pectively . The class of RN compact spaces , on the other hand, do es show other p er manence pr op erties pr esent a lso for Eb erlein compact spaces and man y other classes of compact spaces playing important roles in Bana ch space theory . Namely , there is a n iso morphism inv aria nt clas s of Ba nach spaces (of Asplund generated spac es) asso ciated with it in the sense that if K is an RN compact then, the space C ( K ) o f r eal v alued cont inuous functions on K is an Asplund generated space, and if X is a n Asplund generated space, then the dual ball B X ∗ is RN compact. A version of the ab ov e question on c ontin uous images of RN compacta on the Banach space level, i.e., if subspaces of Asplund generated spaces are Asplund generated was answered in the negative alrea dy 30 years ago in [35]. In this language our r esult is equiv alent to constr ucting a subspace Y ⊆ X of an Asplund ge nerated space X s uc h that the dual ba ll B Y ∗ is not RN co mpact (see [14]). Note that Stegall’s argument from [35] is far from achieving this, as it uses Rosenthal’s non W CG s ubspace of a WCG space from [3 2], but by [8] the dual unit ball of the subspace is e ven an Eb erlein compa ctum and so RN compact. The p oint here is that B X ∗ may b e RN compact for X not Asplund gener ated but B C ( K ) ∗ is RN compact if and only if K is RN compact if and o nly if C ( K ) is Asplund g enerated. W e also hav e a similar chain of equiv alences for RN r eplaced b y a co n tinuous image of RN and Asplund g enerated repla ced by a subspace of Asplund ge nerated (see [12, 14, 15]). It follows that bo th classes of RN compact spaces a nd th eir contin uous images are stable under taking iso morphism of their space of contin uous functions, meaning tha t (1) If L is RN compact and C ( K ) is isomo rphic to C ( L ), then K is also RN compact. (2) If L is a co nt inuous image of an RN compact space and C ( K ) is iso morphic to C ( L ), then K is also a contin uous image of a n RN compact space. Now, if we combine these fac ts with the alre ady mentioned result that an a l- most to tally disc onnected imag e of a n RN compactum is RN compact, we obtain a remark able consequence of o ur exa mple: 3 Corollary 1 . 2. The sp ac e C ( L 1 ) is not isomorphic t o any C ( K ) wher e K is almost total ly disc onne cte d. The questio n whether there could exist a compact space L such that C ( L ) is not isomorphic to any C ( K ) with K to tally disconnected ha s b een a long sta nding op en problem motiv a ted b y the Bessaga Milutin Pe lczy ´ nski classification of sepa- rable Banach spaces of the fo rm C ( K ). It was first solved in the negative by the second a uthor in [21]. Howev er the ex ample obtained there (and others which hav e bee n constructed later with similar techniques like in [29]) is very different from this one, b eca use in that ca se C ( L ) was an indeco mpo sable B anach space. This in particular means , on the level of compact space L , that it contains no c onv er- gent seq uences and is strong ly rigid (all noniden tity contin uous maps fr om L in to itself a re constant) as shown in [33]. Moreover t he dual ball B C ( L ) ∗ with the weak ∗ top ology satisfies a strong rigidity condition (23 of [22]). How ever, our space L 1 has many nontrivial contin uous transfor mations into itself and as a contin uous ima ge of an RN compa ctum, it is sequentially co mpact [2 4] and hence C ( L 1 ) c ontains many infinite-dimensio nal co -infinite-dimensional co mplement ed subspaces. Th us the fact t hat a C ( K ) space is not isomorphic to any C ( L ) for L to tally disconnected do es not imply prop erties of spaces fro m [21] lik e indecomp os ability or not b eing isomorphic to its h yp erplanes and the geometr y of such a spa ce can b e quite nice. Let us now explain the main idea of our constr uction. F or this we ne ed a bunch of definitions. Definition 1. 3. Let K b e a topolo gical space and d : K 2 → R + ∪ { 0 } b e a metric on the set K (no t r elated to the topolo gy on K ). (1) W e say that d fragments K if a nd only if for every ε > 0 a nd e very closed F ⊆ K there is a n op en U ⊆ K s uch that U ∩ F 6 = ∅ a nd diam d ( U ∩ F ) = sup { d ( x, y ) : x, y ∈ U ∩ F } < ε. (2) If K ′ , K ′′ ⊆ K , then d ( K ′ , K ′′ ) = inf { d ( x, y ) : x ∈ K ′ , y ∈ K ′′ } , (3) W e say that d is lo wer s emi con tinuous (l.s.c.) if and only if given distinct x, y ∈ K and 0 < δ < d ( x, y ), there are o pe n U ∋ x a nd V ∋ y such that d ( U, V ) > δ . (4) W e say that d is Reznichenk o if and only if g iven distinct x, y ∈ K there are open U ∋ x and V ∋ y such that d ( U, V ) > 0. F ragmentabilit y w as formally introduced in [2 0] and its relation to RN compacta comes from the fact that every b ounded subset of a dual space with the Ra don- Nikod´ ym pro per ty is fra gmented by the dua l norm [27, 34]. A compa ct space K is an RN compact space if and o nly if there is an l.s.c. metr ic on K which frag men ts K [24]. Co mpact spaces which are f rag men ted b y a Reznic henko metric constitute a sup erclass of RN compa ct s paces, sometimes called strongly fragmentable compact spaces [13, 26], but which c oincides with the class of quas i RN compact spaces int ro duced by Arv anitakis [3] by a result of Namiok a [25] (cf. also [13]). What we need to know ab out quasi RN c ompacta is that the ab ov e mentioned result o f Arv anitakis applies to them, that is , totally disconnected quasi RN co mpacta ar e RN co mpacta [3]. The main insig ht t hat le ads to the construction is to see how to destr oy the l.s.c. prop erty of a metric without destroying the Reznichenk o pr op erty . This is described 4 ANTONIO A VIL ´ ES AND PIOTR K OSZMIDER in Pr op ositions 4.1, 4.3. It is done by a “sma rt” replace men t of some point by the unit interv al a nd can b e in terpre ted a s an op eration of the s o c alled resolutio n o f a top ological space. A central role of this method in top olo gy is claimed in [37] where it is traced back to [16]. It is probably not a coincidence that the spaces constructed in [21] can a lso b e viewed a s obtaine d by versions of r esolutions. W e start with an RN compactum which is simple mo difications of appropriate scattered spa ce of height 3, just to make our resolutions p owerful eno ugh. Then we carefully do as many resolutions of nonisola ted po int s as neces sary to destroy all l.s .c. metrics i.e., to make sur e that the resulted s pace is not RN compact. W e need to predict all these l.s.c. metr ics using a co m binatoria l or a descriptiv e set-theor etic tool. Fina lly it turns out that not o nly the space r emains with a Reznichenk o metric after all these resolutions but a lso its s tandard totally disconnected preimage maint ains through the resolutions a metric which frag men ts it. So, it is enough to use the ab ov e mentioned r esult of Arv anitakis to co nclude that this totally disconnected preimage is RN compact. The structure of the pap er is as follows: In Section 2 w e introduce some basic notation. In Section 3 we present what w e call a b asic sp ac e , the star ting point of our constr uction. In Section 4 we explain how to o btain a surjection π : L 0 − → L 1 like in Theo rem 1 .1 from a basic space. Finally in Sections 5 a nd 6 we provide tw o different wa ys o f constructing a basic space. The firs t one is ba sed on a version of the Ciesielski- Pol compact space [9] and can b e done within ZF C without a dditional axioms. The second constructio n is bas ed on ladder sys tems on ω 1 (see [1], [30]) and assumes ♦ . W e found of in terest to include the constructio n under ♦ as well bec ause it has additional prop er ties, for instance separ able s ubspaces of L 0 and L 1 are metriza ble. The compact spaces that we construc t have weight c but w e do not know if per haps b is the optimal weigh t of a co un terexa mple to the problem. The reader can find in [7] a nd [13] a num ber o f in teresting pro blems on RN compa cta that still remain o pen. F or ex ample, we may mention that it is unknown if every RN compact space is the contin uous image of a zer o-dimensional RN co mpact space , or if it is always homeomo rphic to a subspace of the space of probability mea sures on a sc attered s pace. W e do not know as w ell whether the cla ss o f c ontin uous images of RN compact spaces coincides with that o f quasi RN compact spaces . It would be also interesting to find coun terexa mples to restricted forms of the co n tinuous image problem, like the union o f t wo RN compact spa ces not to b e RN, or the c onv ex h ull of an RN compact s pace not to be RN. 2. Some not a tions By ∆ = 2 N we will denote the Can tor set, the set o f a ll infinite sequences o f 0 ’s and 1’s endow ed with the top ology induced by the metric ρ : ∆ × ∆ − → R given by ρ ( x, y ) = 2 − max { k : x k 6 = y k } By T = 2 <ω we denote the set of all finite se quences of 0 ’s a nd 1 ’s. F or t ∈ T by | t | we denote the cardinality of t , that is, its length. If t = ( t 1 , . . . , t n ) ∈ T and s = ( s 1 , . . . ) ∈ T ∪ ∆, we denote t ⌢ s = ( t 1 , . . . , t n , s 1 , s 2 , . . . ). If t = ( t 1 , t 2 . . . ) ∈ T ∪ ∆ and s = ( s 1 , s 2 , . . . ) ∈ T ∪ ∆, t < s refers to the lexicog raphical o rder, so it means 5 that there exists k s uch tha t t k < s k but t i = s i for i < k . Given s, t ∈ T , we consider the co n tinuous function Γ t s : ∆ − → ∆ defined as: • Γ t s ( z ) = t ⌢ (0 , 0 , 0 , . . . ) if z < s , • Γ t s ( s ⌢ λ ) = t ⌢ λ for every λ ∈ ∆, • Γ t s ( z ) = t ⌢ (1 , 1 , 1 , . . . ) if z > s . The function q : ∆ − → [0 , 1 ] is the standard co n tinuous surjection given by q ( t 1 , t 2 , . . . ) = ∞ X k =1 t k 2 k Notice that q transfers the lexicogra phical order of ∆ to the usual order of [0 , 1], in the s ense that x ≤ y implies that q ( x ) ≤ q ( y ). 3. The st ar ting basic sp a ce W e s hall call a b asic sp ac e a co mpact scatter ed space K which can b e written as K = S n ∈ N A n ∪ B ∪ C satisfying the follo wing pro p erties (1) All points o f A = S n A n are iso lated in K . (2) F or every x ∈ B there exists an infinite set C x ⊂ A such that C x = C x ∪ { x } and moreov er, C x is op en in K . (3) There exists a function ψ : B − → N N such that: Given an y family { X n m : m, n ∈ N } of subsets of A with A n = S m X n m for ev ery n , there exists x ∈ B such that 1 C x ∩ X n ψ ( x )[ n ] is infinite for all n . 4. How to o bt ain the desired continu o us ima ge from a basic sp ace The first step is to cons ider the compact s pace L obtained from the basic space K by substituting each p oint of A by a copy o f the Cantor set ∆. That is, L = ( A × ∆) ∪ B ∪ C A ba sic neighbo rho o d o f a p o in t ( a, t ) is of the fo rm { a } × U where U is a neigh- bo rho o d o f t in ∆. A basic neig h b orho o d of a p oint x ∈ B ∪ C is o f the form (( U ∩ A ) × ∆) ∪ U \ A , where U is a neighbor ho o d of x in K . W e shall us e the co un table set T = 2 <ω instead of N in o rder to describ e the basic s pace K . So we shall write A = S t ∈ T A t instead A = S n ∈ N A n , and the last condition o n our basic space will b e now r ead as: (3’) There exists a function ψ : B − → T T such that: Giv en any family { X t s : s, t ∈ T } of subsets of A w ith A t = S s X t s for every t , there exis ts x ∈ B such that C x ∩ X t ψ ( x )[ t ] is infinite for all t ∈ T . F or every x ∈ B we c onsider a contin uous function g x : L \ { x } − → ∆ defined in the fo llowing way: (1) g x ( y ) = 0 whenever y 6∈ C x × ∆, y 6 = x , (2) g x ( a, z ) = Γ t ψ ( x )[ t ] ( z ) for a ∈ A t ∩ C x , z ∈ ∆. 1 we denote b y ψ ( x )[ n ] the ev aluation on n of the function ψ ( x ) : N − → N 6 ANTONIO A VIL ´ ES AND PIOTR K OSZMIDER W e a lso conside r f x : L \ { x } − → [0 , 1], f x = q ◦ g x . Now, we are in a p osition to define the announced π : L 0 − → L 1 . Let L 0 =  [ u, v ] ∈ L × ∆ B : g x ( u ) = v x for a ll x ∈ B \ { u }  L 1 =  [ u, v ] ∈ L × [0 , 1 ] B : f x ( u ) = v x for a ll x ∈ B \ { u }  π [ u, v ] = [ u, q ( v x ) x ∈ B ] . Notice an imp ortant fa ct a b out the structure of L 0 and L 1 . When u ∈ L \ B , there is a unique p oint of L i of the form [ u , v ]. Ho wev er, when u ∈ B , the set { [ u, v ] ∈ L i } is homeo morphic to ∆ when i = 0 a nd to [0 , 1] when i = 1 , b ecause all co ordinates v x are determined by u as v x = g x ( u ) (or f x ( u )) when x 6 = u , but v u can tak e an y v alue from ∆ (or [0 , 1 ]). In this w ay , we c an think th at we hav e sp litte d each po int of B into a Ca nt or set (or into an interv a l) fol lowing the functions g x (or r esp ectively f x ). Prop ositio n 4.1 . L 0 is RN c omp act. Pr o of. W e check firs t that L 0 is closed in K × ∆ B , hence co mpact. So fix [ u, v ] ∈ K × ∆ B \ L 0 and we find a neig hborho o d of [ u, v ] dis joint from L 0 . Since [ u, v ] 6∈ L 0 , there exists x 6 = u such that g x ( u ) 6 = v x . Let V and W be disjoint op en neighborho o ds in ∆ of g x ( u ) and v x resp ectively . Let U be a neigh b orho o d o f u in L such that g x ( U ) ⊂ V and x 6∈ U . The neighborho od we are lo oking for is ˜ U = { [ u ′ , v ′ ] ∈ L × ∆ B : u ′ ∈ U, v ′ x ∈ W } . Indeed, if [ u ′ , v ′ ] ∈ ˜ U , then x 6 = u ′ since x 6∈ U , but g x ( u ′ ) 6 = v ′ x bec ause g x ( u ′ ) ∈ V while v ′ x ∈ W . Consider the following metric d : L 0 × L 0 − → [0 , 1]: (1) d ([ u, v ] , [ u, v ]) = 0, (2) d ([ u, v ] , [ u, v ′ ]) = ρ ( v u , v ′ u ) if u ∈ B , (3) d ([ u, v ] , [ u ′ , v ′ ]) = ρ ( r , r ′ ) if u, u ′ ∈ A × ∆, u = ( a, r ), u ′ = ( a, r ′ ), (4) d ([ u, v ] , [ u ′ , v ′ ]) = 1 in any remaining case when [ u, v ] 6 = [ u ′ , v ′ ]. Claim 1. The metric d fragments L 0 . Recall the definition of fra gmentabilit y (1) 1.3 and consider a nonempty Y ⊂ L 0 , (1) If Y co n tains a p oint o f the form [ u, v ], with u = ( a, r ) ∈ A × ∆, then ta ke U a neigh b orho o d of r in ∆ of ρ -diameter less than ε , and then V = { [ u, v ] : u = ( a, s ) , s ∈ U } is a neighborho o d of [ u, v ] o f d - diameter les s than ε . (2) If Y does not contain any p oint as in the previo us case, then u ∈ B ∪ C for a ll [ u, v ] ∈ Y . Since B ∪ C ⊂ K is scattered, we can find u 0 an iso lated po int o f the set Z = { u ∈ B ∪ C : ∃ v [ u , v ] ∈ Y } . Suppose [ u 0 , v 0 ] ∈ Y , let U be a neighbor ho o d of u 0 in L that iso lates u 0 inside Z , and W a neighborho o d of v 0 u 0 in ∆ of ρ -diameter le ss than ε . Then V = { [ u, v ] ∈ Y : u ∈ U, v u 0 ∈ W } ⊂ { [ u 0 , v ] : v u 0 ∈ W } is a nonempty relative op en subset of Y of d -diameter le ss than ε . 7 Claim 2. The metric d is a Reznichenk o metric. By (4) of 1.3 to prove that d is Reznichenk o, given [ u 0 , v 0 ] 6 = [ u 1 , v 1 ], w e must find neighborho o ds U a nd V of [ u 0 , v 0 ] and [ u 1 , v 1 ] resp ectively s uch tha t d ( U, V ) = inf { d ( z , z ′ ) : z ∈ U, z ′ ∈ V } > 0 W e dis tinguish several ca ses: (1) If u 0 , u 1 ∈ A × ∆, u 0 = ( a, r ), u 1 = ( a, r ′ ), then we can take J and J ′ neighborho o ds of r and r ′ resp ectively at positive ρ -distance, and then take U = { [( a, s ) , v ] ∈ L 0 : s ∈ J } and V = { [( a, s ) , v ] ∈ L 0 : s ∈ J ′ } . (2) In an y other case when u 0 6 = u 1 , we can take neighborho o ds G and G ′ of u 0 and u 1 such that d ([ u, v ] , [ u ′ , v ′ ]) = 1 whe never u ∈ G and u ′ ∈ G ′ . (3) If u 0 = u 1 = x ∈ B , we consider G a nd G ′ disjoint clop en neighborho o ds of v 0 x and v 1 x resp ectively inside ∆. Let W = ( C x × ∆) ∪ { x } which is a clop en neig hborho o d of x in L . W e claim that U = { [ u, v ] : u ∈ W , v x ∈ G } and V = { [ u , v ] : u ∈ W, v x ∈ G ′ } are at a pos itiv e d -distance as require d. If they were not, we could find sequences e n ∈ U a nd ˜ e n ∈ V such that d ( e n , ˜ e n ) → 0. W e can s uppo se that e n = [( a n , z n ) , v n ], ˜ e n = [( a n , ˜ z n ) , ˜ v n )], the a n is the sa me in both ca ses s ince otherwise d ( e n , ˜ e n ) = 1. By pas sing to a subsequence we ca n supp ose tha t v n x → w ∈ G and ˜ v n x → ˜ w ∈ G ′ . By pas sing to a further subsequence, we ca n reduce this case to one o f the following tw o sub cases : (a) either there is a t such that a n ∈ A t for all n . Then v n x = Γ t ψ ( x )[ t ] ( z n ) and ˜ v n x = Γ t ψ ( x )[ t ] ( ˜ z n ). Since ρ ( z n , ˜ z n ) = d ( e n , ˜ e n ) → 0, the co nt inuit y of Γ t ψ ( x )[ t ] implies that ρ ( w , ˜ w ) = 0, a co n tradiction. (b) o r a n ∈ A t n and | t n | → ∞ . In that case, the ρ -dia meter of Γ t n ψ ( x )[ t n ] (∆) = { t ⌢ n λ : λ ∈ ∆ } tends to 0 as well. Again, this implies that ρ ( v n x , ˜ v n x ) → 0 a nd w = ˜ w , a con tradictio n. Every compact space frag men ted b y a Reznichenk o metric is quasi- RN [25], and every zer o-dimensiona l quasi-RN compac t s pace is RN c ompact [3 ].  Remark 4.2. Ther e is a quite natural wa y of r edefining the metr ic d on the pa irs [ u, v ] , [ u, v ′ ] for u ∈ B and v ∈ ∆ to obtain an l.s.c. quas imetric (see [3]) on L 0 which c ould give ano ther pro of o f the RN pr op erty following the results of [3] Prop ositio n 4.3 . L 1 is not RN c omp act. Pr o of. Fir st, L 1 is compact being a contin uous image of L 0 . If L 1 is RN compa ct, then there exists a low er semicontin uous metric δ : L 1 × L 1 − → R which fragments L 1 . Giv en a ∈ A and z ∈ ∆ let us denote by a + z the unique p oint o f L 1 of the form a + z = [( a, z ) , v ]. B y the frag ment ability co ndition, whenever a ∈ A t we ca n find s ( a ) ∈ T such that δ ( a + s ( a ) ⌢ (0 , 0 , . . . ) , a + s ( a ) ⌢ (1 , 1 , . . . )) < 1 4 | t | . Let X t s = { a ∈ A t : s ( a ) = s } , so tha t A t = S s ∈ T X t s for every t ∈ T . W e are in the po sition to apply the fundamen tal pr op erty (3 ’) o f our basic space, so that w e can find x ∈ B such that C x ∩ X t ψ ( x )[ t ] is infinite for all t ∈ T . This mea ns that for every 8 ANTONIO A VIL ´ ES AND PIOTR K OSZMIDER t ∈ T we can find an infinite sequence { a n } ⊂ C x ∩ A t such that s ( a n ) = ψ ( x )[ t ] for every n . Now, for every ξ ∈ [0 , 1] let us denote x ⊕ ξ = [ x, v ] ∈ L 1 , where v x = ξ and v y = f y ( x ) for y ∈ B \ { x } . If we remember the definition of f x and g x , we notice that f x ( a n + ψ ( x )[ t ] ⌢ (0 , 0 , 0 , . . . )) = q ( t ⌢ (0 , 0 , 0 , . . . )) =: t 0 f x ( a n + ψ ( x )[ t ] ⌢ (1 , 1 , 1 , . . . )) = q ( t ⌢ (1 , 1 , 1 , . . . )) =: t 1 Now, taking limits when n → ∞ , a n + ψ ( x )[ t ] ⌢ ( i, i, i, . . . ) → x ⊕ ξ i where, by lo oking at the x -co ordinate, ξ i = lim n f x ( a n + ψ ( x )[ t ] ⌢ ( i, i, i, . . . )) = q ( t ⌢ ( i, i, i, . . . )) = t i Using the low er semico n tinuit y 2 of δ , we co nclude that δ ( x ⊕ t 0 , x ⊕ t 1 ) ≤ 1 4 | t | and this happ ens for ev ery t ∈ T . Now fix m ∈ N , and observe that  [ t 0 , t 1 ] : t ∈ T , | t | = m  =  [( k − 1)2 − m , k 2 − m ] : k = 1 , . . . , 2 m  so we can apply the triang le inequality of the metric δ and w e o btain that δ ( x ⊕ 0 , x ⊕ 1) ≤ 2 m 1 4 m = 1 2 m but this happ ening for every m contradicts the fact that δ ( x ⊕ 0 , x ⊕ 1 ) > 0.  Remark 4.4. Note that the ab ov e pro o f do es not work for L 0 bec ause { [ t ⌢ (0 , 0 , 0 , . . . ) , t ⌢ (1 , 1 , 1 , . . . )] : t ∈ T , | t | = m } do not form c onsecutive in terv als, that is, their left ends a re not equal to any of their right ends, and s o the tr iangle inequality canno t b e applied as in the pro of ab ov e. 5. A basic sp ace of the f orm of the Ciesielski-Pol comp act Remem b er that a set S ⊂ R is called a Bernstein set if bo th S ∩ P a nd S \ P are nonempty fo r every p erfect set P ⊂ R . The classical result of Bernstein is tha t such a set exis t: it is constructed b y trans finite induction by enumerating a ll p os- sible p erfect s ubsets of R as { P ξ : ξ < c } and a t every step ξ choos ing new p oints x ξ , y ξ ∈ P ξ and declaring x ξ ∈ S and y ξ 6∈ S . A mino r mo difica tion of this ar gu- men t yields the existence of c man y disjoint Bernstein sets: write c = S { I α : α < c } with | I α | = c , a nd assume that fo r every α , { P ξ : ξ ∈ I α } enumerates all p erfect subsets of R ; then at step ξ ∈ I α , choose new x ξ , y ξ ∈ P ξ and declare x ξ ∈ S α and y ξ 6∈ S α . The basic space that we are g oing to co nstruct is of the form K = S n ∈ N A n ∪ B ∪ {∞} where the sets A n and the set B a re pairwise disjoin t B ernstein s ubsets of R . All p oints of A = S n ∈ N A n will b e of course iso lated, the spa ce A ∪ B will b e lo cally compact (its top ology will b e a refinement of the to po logy inherited from R ) a nd K its o ne-p oint compactification. In order to desc rib e co mpletely o ur basic space we need to say which ar e the sets C x for x ∈ B (that will provide a basis 2 W e are using the fol lowing prop erty of a l o wer semicont inuous metric, which i s a direct consequenc e of Definition 1.3: if x n → x , y n → y and δ ( x n , y n ) ≤ ε for every n , then δ ( x, y ) ≤ ε . 9 of neighbo rho o ds of such x ∈ B : all H ∪ { x } where H is co finite in C x ) a nd also which is the function ψ : B − → N . All top ological notions b elow r efer to the standard topo logy on R . Let ( F α ) α< c be an enumeration of all s equences ( F α ( n, m )) n,m ∈ N of co un table subs ets o f R such that F α ( n, m ) ⊆ A n for ea ch n, m ∈ N a nd ⊞ F α = \ n ∈ N [ m ∈ N F α ( n, m ) contains a p erfect set. W e constr uct { x α : α < c } ⊆ B , the sets C x α and and ψ ( x α ) by induction on α < c . Given α we pick x α ∈ ⊞ F α \ { x β : β < α } . W e define ψ ( x α ) = ( m n ) n ∈ N to b e suc h a sequence that x α ∈ T n ∈ N F α ( n, m n ) which exists s ince x α ∈ ⊞ F α . Then take as C x α the ter ms of a sequence which c onv erges in R to x α and such tha t for every n , C x α contains infinitely man y elements from F α ( n, m n ). After the inductive pr o cedure is finished, F or the remaining elements x ∈ B \ { x α : α < c } , w e define ψ ( x ) to b e any arbitra ry v alue, and C x any s equence of elements o f A conv ergent to x . W e fina lly chec k that the key prop erty (3) of ba sic spaces is satisfied. Supp ose that we hav e A n = S X n m for e very n . F or every n , the set S X n m is a Borel s et and it intersects every p erfect set since it co n tains the Bernstein set A n . Therefor e S X n m is co countable in R (every uncountable B orel set contains a p erfect set). Therefore \ n ∈ N [ m ∈ N X n m is co countable and in particular, contains a per fect set. W e choo se count able sets F ( n, m ) ⊂ X n m with F ( n, m ) = X n m . Then, since ⊞ F α = \ n ∈ N [ m ∈ N X n m contains a per fect set, this sequence mu st app ear in o ur enumeration as F = F α for some α < c . Let us see that x = x α is the element of B that we are lo oking for . Indeed, by the way we chose C x α and ψ ( x α ), we know that C x α contains infinitely many elements from F α ( n, m n ) ⊂ X n m n = X n ψ ( x α )[ n ] , for ev ery n . This finishes the pro of. 6. A basic sp ace fr om a ladder system under ♦ Definition 6.1. ( D α ) α ∈ ω 1 is ca lled a ♦ -sequence if and only if for every X ⊆ ω 1 the s et { α ∈ ω 1 : D α = X ∩ α } is stationary (i.e., intersects all closed in the order top ology and unbounded subsets of ω 1 ). ♦ is a statement that a ♦ -sequence exis ts. In the basic space that we constr uct now, K = S ∞ n =1 A n ∪ B ∪ C , we will ha ve (1) A n is the set o f all co un table ordinals of the form α + n , where α is a limit ordinal. Hence A = S ∞ 1 A n is the set of all countable success or or dinals. (2) B is the set of all co un table limit or dinals, ex cept 0. 10 ANTONIO A VIL ´ ES AND PIOTR K OSZMIDER (3) C = { ω 1 } . The sets { C x : x ∈ B } will b e a la dder system in ω 1 . That is, for every x ∈ B , C x = { β 1 , β 2 , . . . } ⊂ A with β 1 < β 2 < · · · and sup { β n : n < ω } = x . Once the ladder system is given, the top ology cons idered o n K is such that each p o in t of A is iso lated, a basis of neig hborho o ds o f x ∈ B are the sets H ∪ { x } with H co finite in C x , and K is the one-p oint compactification of the lo cally compact space A ∪ B . Now, we have to explain how to find a ladder system { C x : x ∈ B } and a f unction ψ : B − → N N so tha t the fundamen tal prop erty (3) of a basic space is satisfied. So let ( D α ) α<ω 1 be a ♦ -sequence. Let x ∈ B . Supp ose first that: (1) x 6 = ω , (2) D x ⊂ A , (3) sup( D x ∩ A n ) = x fo r every n , and (4) there exists f : N → N such that D x ∩ ω = { 2 n 3 f ( n ) : n = 1 , 2 , 3 , . . . } . If a ll this co nditions ho ld, we define ψ ( x ) = f and C x to b e some increasing se- quence of elements of D x \ ω whose s upremum is x a nd which cont ains infinitely many elements o f A n for every n . F or the remaining x ∈ B tha t do not s atisfy the conditions above, we define C x and ψ ( x ) in an a rbitrary w ay . Now supp ose that A n = S m X n m as in condition (3) of a basic space. F or every n , choose m n ∈ N such that X n m n is unco un table. Let G n be the set of all limit ordinals which a re the supremum o f some sequence c ontained in X n m n . Notice that this is a closed and unbounded subset o f ω 1 . Define X = { 2 n 3 m n : 1 ≤ n < ω } ∪ ∞ [ n =1 ( X n m n \ ω ) . By the choice of ( D α ) α ∈ L ( ω 1 ) , there is α > ω , α ∈ T ∞ n =1 G n such that X ∩ α = D α . 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Recen t progress in general topology (Prague, 1991), 673 - 757, North-Holland, Amsterdam, 1992. 12 ANTONIO A VIL ´ ES AND PIOTR K OSZMIDER E-mail addr ess : avileslo@um .es Dep ar t amen to de Ma tem ´ aticas, Universidad de Murcia, 30100 Murcia (Sp ain) E-mail addr ess : P.Koszmider @Impan.pl Institute of Ma thema tics, Polish Academy of Sciences, ul. ´ Sniadeckich 8, 00 -9 56 W arsza w a, Poland